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Communications in Theoretical Physics

Comparison between the effects of Hamiltonian and Liouvillian exceptional points on the Leggett–Garg inequality and no-signaling-in-time conditions

Kun-Jie Zhou, Jian Zou, Bao-Ming Xu, Bin Shao

Published Feb 13, 2025• 0 citations
Publisher

Abstract

Abstract In this paper, we investigate the effect of exceptional points (EPs) on the violation of Leggett–Garg inequality (LGI) and no-signaling-in-time (NSIT) conditions and compare the different effects between the Hamiltonian EP (HEP) and Liouvillian EP (LEP) on those violations. We consider an open system consisting of two coupled qubits and each qubit is contacted with a thermal bath at a different temperature. In the case of omitting quantum jumps, we find that the system exhibits a second-order HEP, which separates the parameter space into an overdamped regime and an underdamped regime. In this situation, the LGI and NSIT conditions can be violated in both regimes and not violated at the HEP. In the case of without omitting quantum jumps, we find that the system exhibits a third-order LEP, which also separates the parameter space into an overdamped regime and an underdamped regime. In this situation, the LGI can only be violated in the underdamped regime with large coupling strength between the qubits. Conversely, the NSIT conditions can be violated in both regimes, as well as at the LEP, except in the overdamped regime with small coupling strength between the qubits. Comparing the violations of the LGI and NSIT conditions with HEP and LEP, we find that the quantum jumps would reduce the generation of coherence, enhance the decoherence, and lead to narrower parameter regimes that the LGI and NSIT conditions can be violated. Furthermore, in both cases, the NSIT conditions can be violated in a wider parameter regime than the LGI.

DOI
10.1088/1572-9494/adb562

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Abstract In this paper, we investigate the effect of exceptional points (EPs) on the violation of Leggett–Garg inequality (LGI) and no-signaling-in-time (NSIT) conditions and compare the different effects between the Hamiltonian EP (HEP) and Liouvillian EP (LEP) on those violations. We consider an open system consisting of two coupled qubits and each qubit is contacted with a thermal bath at a different temperature. In the case of omitting quantum jumps, we find that the system exhibits a second-order HEP, which separates the parameter space into an overdamped regime and an underdamped regime. In this situation, the LGI and NSIT conditions can be violated in both regimes and not violated at the HEP. In the case of without omitting quantum jumps, we find that the system exhibits a third-order LEP, which also separates the parameter space into an overdamped regime and an underdamped regime. In this situation, the LGI can only be violated in the underdamped regime with large coupling strength between the qubits. Conversely, the NSIT conditions can be violated in both regimes, as well as at the LEP, except in the overdamped regime with small coupling strength between the qubits. Comparing the violations of the LGI and NSIT conditions with HEP and LEP, we find that the quantum jumps would reduce the generation of coherence, enhance the decoherence, and lead to narrower parameter regimes that the LGI and NSIT conditions can be violated. Furthermore, in both cases, the NSIT conditions can be violated in a wider parameter regime than the LGI.
Abstract Leggett Garg inequalities (LGIs) provides an elegant way for probing the incompatibility between the notion of macrorealism and quantum mechanics. For unitary dynamics the optimal quantum violation of a LGI is constrained by the L u ¨ ders bound. In this paper, we have studied two formulations of LGIs in three-time LG scenario, viz, the standard LGIs and third-order LGIs both for unbiased and biased measurement settings. We show that if system evolves under non-unitary quantum channel between two measurements, the quantum violations of both forms of LGIs exceed their respective L u ¨ ders bounds and can even reach their algebraic maximum in sharp measurement settings. We found that when the measurement is unsharp the quantum violations of both standard and third-order LGIs for non-unitary quantum channel can be obtained for lower value of the unsharpness parameter compared to the unitary dynamics. We critically examined the violation of L u ¨ ders bound of LGIs and its relation to the violation of various no-signaling in time conditions, another formalism for testing macrorealism. It is shown that mere violations of no-signaling conditions are not enough to warrant the violation of standard LGIs, an interplay between the violations of various NSIT condition along with a threshold value play an important role. On the other hand violation of third-order LGI is obtained when the degree of violation of a specific no-signaling in time condition reaches a different threshold value.
Violations of the Leggett-Garg inequality (LGI) up to its algebraic bound under non-Hermitian dynamics are well established theoretically. Here, we demonstrate that such extreme violations are intrinsically fragile when realistic measurement processes are taken into account. We consider an open two-level system described by a time-local hybrid Liouvillian, with a continuous parameter q∈[0,1]q \in [0,1]q∈[0,1], representing detector efficiency, i.e., the fraction of quantum jump trajectories that are retained in the ensemble. This parameter interpolates between trace-preserving Lindblad dynamics (q=1q=1q=1) and non-Hermitian ``no-jump" evolution (q=0q=0q=0). While K3K_3K3​ approaches its algebraic maximum of 3 in the null-efficiency limit, even an infinitesimal increase in detector efficiency induces a rapid, highly nonlinear suppression toward the classical bound. This logarithmic sensitivity reveals that maximal LGI violations are not robust physical features but rather singular limits of idealized measurement conditions. Our results have direct experimental implications: achieving algebraic LGI violations in systems undergoing continuous time evolution requires near-perfect suppression of detected quantum jumps (i.e., effective post-selection), placing stringent constraints on detector performance. In contrast to discrete protocols based on time-non-divisible dynamics, our framework shows that extreme violations arising within continuous, divisible quantum trajectory evolution constitute a fundamentally fragile regime.
We investigate both analytically and numerically violations of a Leggett-Garg inequality (LGI) for a composite quantum system in contact with two separate reservoirs at different temperatures. Remarkably we find that LGI violations can be enhanced when a heat current is established at low temperatures in a steady-state regime. Based on a Kraus operator decomposition of the nonunitary evolution for a system formed by two interacting spins or quantum bits, we provide analytical support for power-law relations between dissipation strength and mean temperature in the borderlines separating parameter regions where nonequilibrium conditions affect differently the maximal LGI violation. Furthermore, a correspondence between spatial and temporal correlation inequalities is shown to persist even in such nonequilibrium thermal settings.
Abstract We report an unusual buildup of the quantum coherence in a qubit subjected to non-Hermitian evolution generated by a parity-time ( PT ) symmetric Hamiltonian, which is reinterpreted as a Hermitian system in a higher dimensional space using Naimark dilation. The coherence is found to be maximum about the exceptional points (EPs), i.e. the points of coalescence of the eigenvalues as well as the eigenvectors. The nontrivial physics about EPs has been observed in various systems, particularly in photonic systems. As a consequence of enhancement in coherence, the various formulations of Leggett–Garg inequality tests show maximal violation about the EPs.
The Leggett-Garg inequality (LGI) delineates a boundary between quantum and classical systems. While the temporal quantum correlations represented by LGIs have been extensively investigated in the Hermitian domain, the exploration of LGIs under non-Hermitian conditions remains scarce. Theoretical conjectures posit that nonunitary dynamics could transcend the boundary of LGIs set by Hermitian quantum mechanics, yet its empirical validation is lacking. Here, we demonstrate the enhanced violation of LGIs in a non-Hermitian system of a parity-time (PT\mathcal{PT}PT)-symmetric trapped-ion qubit. The upper bounds of both the third-order parameter K3{K}_{3}K3​ and fourth-order parameter K4{K}_{4}K4​ increase with dissipation and can reach the maximum value when the system approaches the exceptional point, where ${K}_{3}={C}_{21}+{C}_{32}\ensuremath{-}{C}_{31}$ is comprising three two-time correlations and ${K}_{4}={C}_{21}+{C}_{32}+{C}_{43}\ensuremath{-}{C}_{41}$ is comprising four correlations. We also find that the lower bounds of K3{K}_{3}K3​ and K4{K}_{4}K4​ exhibit distinctive behaviors, where the lower bound of K3{K}_{3}K3​ remains constant, but that of K4{K}_{4}K4​ depends on the target states and measurement operators. These findings reveal a pronounced correlation between the dissipative nature of a quantum system and its temporal correlations.
In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility between macrorealism and quantum mechanics. In this paper, we point out that, based on the same assumptions of macrorealism that are used to derive Leggett-Garg inequalities (LGIs), there is a scope of formulating another class of inequalities different from the standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of the standard LGIs and show that its quantum violation is larger than the standard LGIs. By extending this formulation to the nnn-time measurement scenario, we found that the quantum violations of variants of the LGIs for a qubit system increase with nnn, and, for a sufficiently large nnn, an algebraic maximum can be reached. We then compare the violations of the standard and variants of the LGIs in an unsharp measurement scenario and show that, for any arbitrary nnn, the violation of the latter is more robust to unsharpness than the former. Furthermore, we examine the relation between the quantum violations of the variants of the LGIs and an another formulation macrorealism, known as no-signaling in time conditions.